Iterative phase contrast CT reconstruction with novel tomographic operator and data-driven prior

Breast cancer remains the most prevalent malignancy in women in many countries around the world, thus calling for better imaging technologies to improve screening and diagnosis. Grating interferometry (GI)-based phase contrast X-ray CT is a promising technique which could make the transition to clinical practice and improve breast cancer diagnosis by combining the high three-dimensional resolution of conventional CT with higher soft-tissue contrast. Unfortunately though, obtaining high-quality images is challenging. Grating fabrication defects and photon starvation lead to high noise amplitudes in the measured data. Moreover, the highly ill-conditioned differential nature of the GI-CT forward operator renders the inversion from corrupted data even more cumbersome. In this paper, we propose a novel regularized iterative reconstruction algorithm with an improved tomographic operator and a powerful data-driven regularizer to tackle this challenging inverse problem. Our algorithm combines the L-BFGS optimization scheme with a data-driven prior parameterized by a deep neural network. Importantly, we propose a novel regularization strategy to ensure that the trained network is non-expansive, which is critical for the convergence and stability analysis we provide. We empirically show that the proposed method achieves high quality images, both on simulated data as well as on real measurements.

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-Due to legal restrictions, we are not allowed to share the patient data obtained with written informed consent under the ethical approval KEK-2012 554 granted by the Cantonal Ethics Commission of canton Zürich. We can however share our in-silico breast phantoms and the corresponding sinogram data. They will be uploaded to the ETH research collection archive as supporting material to our paper: https://www.research-collection.ethz.ch. The exact URL will follow.

Reviewers' comments:
The compared baseline methods only include conventional methods. The literature has shown that the deep learning has shown better/competitive performance than iterative algorithms in some applications. Therefore, deep learning-based methods, such as postprocessing FBP reconstructed images, should be considered as a competitive baseline.
-We added the comparison to a deep learning-based post-processing step in the "Results" section and in the "Effect of the proposed data-driven regularizer" section.
For the comparison to be fair, we used the same network as within the iterative reconstruction. We would like to stress that this comparison only makes sense when it is possible to analytically reconstruct the phase contrast volumes (which is not always the case, e.g. in Teuffenbach et al., 2017). -We accordingly edited Figs 9 and 10. I am wondering if the authors could provide the comparison of reconstruction times, which is a factor for clinical use.
-We added a paragraph in the "Results" section describing the overall reconstruction times of the two iterative methods, as well as a more detailed description of the computational times of the single steps of the algorithms.
In literature there are many works synergizing deep learning and iterative methods. A brief discussion on key references may be important to further highlight the novelty of the proposed methods.
-We added a paragraph in the "Contributions" section that discusses prior work on combining deep learning with iterative methods and explained where the novelty of our method lies and why it is relevant to the scientific community.

I am not convinced the proposed approach can be called data-driven as the denoising part is associated with the model space (the images), not the data space;
-In the machine learning community, the term "data-driven" refers to algorithms that are fitted on training data, to differentiate them from algorithms that are solely designed by human engineering, and which do not contain trainable parameters. The fact that the mapping is applied in image space does not preclude the term datadriven in our opinion.
In both the synthetic and the experimental tests, the proposed approach tends to remove many details (that are spatially consistent, so, probably not related to random noise, and that are connecting different subdomains in the reconstruction). My impression is that similar results could be obtained, for example, with a much simpler spatial filter or a more appropriate choice of the regularization within the framework of more "standard" approaches. This, of course, does not mean that the proposed approach is not interesting, but a more accurate and fair comparison with the "mainstream" approaches would make the paper more relevant to the readers.
-The article already contains a comparison to the most celebrated and successful classical regularization scheme for CT, i.e. TV regularization. As it can be seen from the results in Figs 9 and 10 and in Table 1, TV regularization is not able to achieve comparable performance to the data-driven PnP strategy. -Moreover, the new comparison with FBP-denoising uses a CNN that is composed of spatial filters, which can thus be regarded as a spatial filter. -It is true that some small details are lost during the denoising. However, given the high amount of noise in the data and the very small size of the structures, we believe it is unrealistic to hope for those small features to be recovered.
A more in-depth discussion of the noise in the data would be important. Moreover, why no data covariance matrix is included in the inversion algorithm? It seems to me that the proposed strategy might be affected, for example, by severe modeling error that is not taken into account and that might lead to dangerous over-interpretation of the results. I do not expect the authors to modify their algorithm accordingly (at least for this manuscript), but I feel a short discussion about that would be important.
-We added a paragraph in the "DPC forward and backward tomographic operators" section in which we argue why we did assume constant variance in the data. The reason is that to accurately estimate the DPC variance, one needs to know the darkfield signal. Since the dark-field signal is hard to accurately compute based on highly noisy data, we decided not to include this into our model. We are currently working on a new algorithm which explicitly models the variance in the intensity data instead of the retrieved DPC data.
P.S. Authors must improve the quality of the images.
-All images were very high-resolution and all passed the PACE system test. Please let us know in case we must edit them.